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<h2 id="matlab绘制爱心">Matlab绘制爱心 </h2>
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<h2 id="matlab绘制爱心-1">Matlab绘制爱心 </h2>
<p>下面给出这个的实现。首先给出桃心形线的参数方程：<br>
<span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mtable rowspacing="0.25em" columnalign="center" columnspacing="0em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mi>x</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo><mn>4</mn><msup><mrow><mi>sin</mi><mo>⁡</mo></mrow><mn>3</mn></msup><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo><mn>12</mn><mi>sin</mi><mo>⁡</mo><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>−</mo><mn>4</mn><mi>sin</mi><mo>⁡</mo><mo stretchy="false">(</mo><mn>3</mn><mi>t</mi><mo stretchy="false">)</mo></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mi>y</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo><mn>13</mn><mi>cos</mi><mo>⁡</mo><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>−</mo><mn>5</mn><mi>cos</mi><mo>⁡</mo><mo stretchy="false">(</mo><mn>2</mn><mi>t</mi><mo stretchy="false">)</mo><mo>−</mo><mn>2</mn><mi>cos</mi><mo>⁡</mo><mo stretchy="false">(</mo><mn>3</mn><mi>t</mi><mo stretchy="false">)</mo><mo>−</mo><mi>cos</mi><mo>⁡</mo><mo stretchy="false">(</mo><mn>4</mn><mi>t</mi><mo stretchy="false">)</mo></mrow></mstyle></mtd></mtr></mtable><annotation encoding="application/x-tex">\begin{gather*} 
x(t) = 4\sin^3(t) = 12\sin(t) - 4\sin(3t) \\ 
y(t) = 13\cos(t) - 5\cos(2t) - 2\cos(3t) - \cos(4t) 
\end{gather*}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:3.0319em;vertical-align:-1.2659em;"></span><span class="mord"><span class="mtable"><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.7659em;"><span style="top:-3.8941em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="mopen">(</span><span class="mord mathnormal">t</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord">4</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop"><span class="mop">sin</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8719em;"><span style="top:-3.1208em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">3</span></span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal">t</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord">12</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop">sin</span><span class="mopen">(</span><span class="mord mathnormal">t</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord">4</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop">sin</span><span class="mopen">(</span><span class="mord">3</span><span class="mord mathnormal">t</span><span class="mclose">)</span></span></span><span style="top:-2.3941em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mopen">(</span><span class="mord mathnormal">t</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord">13</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop">cos</span><span class="mopen">(</span><span class="mord mathnormal">t</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord">5</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop">cos</span><span class="mopen">(</span><span class="mord">2</span><span class="mord mathnormal">t</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord">2</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop">cos</span><span class="mopen">(</span><span class="mord">3</span><span class="mord mathnormal">t</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mop">cos</span><span class="mopen">(</span><span class="mord">4</span><span class="mord mathnormal">t</span><span class="mclose">)</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.2659em;"><span></span></span></span></span></span></span></span></span></span></span></span></p>
<p>那么任一点的位置就可以用复函数 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>z</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo><mi>x</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>+</mo><mi>i</mi><mi>y</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">z(t) = x(t) + iy(t)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.04398em;">z</span><span class="mopen">(</span><span class="mord mathnormal">t</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">x</span><span class="mopen">(</span><span class="mord mathnormal">t</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">i</span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mopen">(</span><span class="mord mathnormal">t</span><span class="mclose">)</span></span></span></span> 来表示。对 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>z</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">z(t)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.04398em;">z</span><span class="mopen">(</span><span class="mord mathnormal">t</span><span class="mclose">)</span></span></span></span> 求傅立叶级数展开 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>z</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo><msubsup><mo>∑</mo><mrow><mi>k</mi><mo>=</mo><mo>−</mo><mi mathvariant="normal">∞</mi></mrow><mrow><mo>+</mo><mi mathvariant="normal">∞</mi></mrow></msubsup><msub><mi>a</mi><mi>k</mi></msub><msup><mi>e</mi><mrow><mi>i</mi><mi>k</mi><mi>t</mi></mrow></msup></mrow><annotation encoding="application/x-tex">z(t) = \sum_{k = - \infty}^{+\infty} a_k e^{ikt}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.04398em;">z</span><span class="mopen">(</span><span class="mord mathnormal">t</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.2693em;vertical-align:-0.358em;"></span><span class="mop"><span class="mop op-symbol small-op" style="position:relative;top:0em;">∑</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.9112em;"><span style="top:-2.4003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span><span class="mrel mtight">=</span><span class="mord mtight">−</span><span class="mord mtight">∞</span></span></span></span><span style="top:-3.2029em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">+</span><span class="mord mtight">∞</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.358em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal">e</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8491em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">ik</span><span class="mord mathnormal mtight">t</span></span></span></span></span></span></span></span></span></span></span></span> 就可以了。</p>
<p>一般地，一个闭合曲线总能找到一个周期的参数方程，所以都可以用这种方法。但对于这个问题，还有一个更讨巧的方法。</p>
<p>注意到 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>x</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">x(t)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">x</span><span class="mopen">(</span><span class="mord mathnormal">t</span><span class="mclose">)</span></span></span></span> 已经表示为正弦级数的形式， <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>y</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">y(t)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mopen">(</span><span class="mord mathnormal">t</span><span class="mclose">)</span></span></span></span> 已经表示为余弦级数的形式，考虑到各个频率分量相互独立以及正弦函数和余弦函数的奇偶性，对每个频率分量：<br>
<span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mtable rowspacing="0.25em" columnalign="center" columnspacing="0em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><msubsup><mi>a</mi><mi>k</mi><mo>+</mo></msubsup><mi>sin</mi><mo>⁡</mo><mo stretchy="false">(</mo><mi>k</mi><mi>t</mi><mo stretchy="false">)</mo><mo>+</mo><msubsup><mi>a</mi><mi>k</mi><mo>−</mo></msubsup><mi>sin</mi><mo>⁡</mo><mo stretchy="false">(</mo><mo>−</mo><mi>k</mi><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo><msubsup><mi>a</mi><mi>k</mi><mi>x</mi></msubsup><mi>sin</mi><mo>⁡</mo><mo stretchy="false">(</mo><mi>k</mi><mi>t</mi><mo stretchy="false">)</mo></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><msubsup><mi>a</mi><mi>k</mi><mo>+</mo></msubsup><mi>cos</mi><mo>⁡</mo><mo stretchy="false">(</mo><mi>k</mi><mi>t</mi><mo stretchy="false">)</mo><mo>+</mo><msubsup><mi>a</mi><mi>k</mi><mo>−</mo></msubsup><mi>cos</mi><mo>⁡</mo><mo stretchy="false">(</mo><mo>−</mo><mi>k</mi><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo><msubsup><mi>a</mi><mi>k</mi><mi>y</mi></msubsup><mi>cos</mi><mo>⁡</mo><mo stretchy="false">(</mo><mi>k</mi><mi>t</mi><mo stretchy="false">)</mo></mrow></mstyle></mtd></mtr></mtable><annotation encoding="application/x-tex">\begin{gather*} 
a_k^+\sin(kt) + a_k^-\sin(-kt) = a_k^x\sin(kt) \\ 
a_k^+\cos(kt) + a_k^-\cos(-kt) = a_k^y\cos(kt) 
\end{gather*}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:3em;vertical-align:-1.25em;"></span><span class="mord"><span class="mtable"><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.75em;"><span style="top:-3.91em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8213em;"><span style="top:-2.4086em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mbin mtight">+</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2914em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop">sin</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="mord mathnormal">t</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8213em;"><span style="top:-2.4086em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mbin mtight">−</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2914em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop">sin</span><span class="mopen">(</span><span class="mord">−</span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="mord mathnormal">t</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7144em;"><span style="top:-2.453em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">x</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop">sin</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="mord mathnormal">t</span><span class="mclose">)</span></span></span><span style="top:-2.41em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8213em;"><span style="top:-2.4086em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mbin mtight">+</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2914em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop">cos</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="mord mathnormal">t</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8213em;"><span style="top:-2.4086em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mbin mtight">−</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2914em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop">cos</span><span class="mopen">(</span><span class="mord">−</span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="mord mathnormal">t</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7823em;"><span style="top:-2.3987em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span><span style="top:-3.1809em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">y</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3013em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop">cos</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="mord mathnormal">t</span><span class="mclose">)</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.25em;"><span></span></span></span></span></span></span></span></span></span></span></span></p>
<p>因此<br>
<span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mtable rowspacing="0.25em" columnalign="center" columnspacing="0em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><msubsup><mi>a</mi><mi>k</mi><mo>+</mo></msubsup><mo>=</mo><mfrac><mrow><msubsup><mi>a</mi><mi>k</mi><mi>y</mi></msubsup><mo>+</mo><msubsup><mi>a</mi><mi>k</mi><mi>x</mi></msubsup></mrow><mn>2</mn></mfrac></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><msubsup><mi>a</mi><mi>k</mi><mo>−</mo></msubsup><mo>=</mo><mfrac><mrow><msubsup><mi>a</mi><mi>k</mi><mi>y</mi></msubsup><mo>−</mo><msubsup><mi>a</mi><mi>k</mi><mi>x</mi></msubsup></mrow><mn>2</mn></mfrac></mrow></mstyle></mtd></mtr></mtable><annotation encoding="application/x-tex">\begin{gather*} 
a_k^+ = \frac{a_k^y + a_k^x}{2} \\ 
a_k^- = \frac{a_k^y - a_k^x}{2} 
\end{gather*}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:4.9192em;vertical-align:-2.2096em;"></span><span class="mord"><span class="mtable"><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.7096em;"><span style="top:-4.7096em;"><span class="pstrut" style="height:3.4736em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8213em;"><span style="top:-2.4086em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mbin mtight">+</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2914em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.4736em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">2</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.6913em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7823em;"><span style="top:-2.3987em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span><span style="top:-3.1809em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">y</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3013em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6644em;"><span style="top:-2.4169em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">x</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2831em;"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span><span style="top:-2.25em;"><span class="pstrut" style="height:3.4736em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8213em;"><span style="top:-2.4086em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mbin mtight">−</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2914em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.4736em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">2</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.6913em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7823em;"><span style="top:-2.3987em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span><span style="top:-3.1809em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">y</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3013em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6644em;"><span style="top:-2.4169em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">x</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2831em;"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:2.2096em;"><span></span></span></span></span></span></span></span></span></span></span></span></p>
<p>这样就求出了正频率和负频率的振幅。</p>
<p><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mtable rowspacing="0.25em" columnalign="center" columnspacing="0em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mi>z</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo><munderover><mo>∑</mo><mrow><mi>k</mi><mo>=</mo><mn>1</mn></mrow><mn>4</mn></munderover><msubsup><mi>a</mi><mi>k</mi><mo>+</mo></msubsup><msup><mi>e</mi><mrow><mi>i</mi><mi>k</mi><mi>t</mi></mrow></msup><mo>+</mo><msubsup><mi>a</mi><mi>k</mi><mo>−</mo></msubsup><msup><mi>e</mi><mrow><mo>−</mo><mi>i</mi><mi>k</mi><mi>t</mi></mrow></msup></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mi>x</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo><munderover><mo>∑</mo><mrow><mi>k</mi><mo>=</mo><mn>1</mn></mrow><mn>4</mn></munderover><msubsup><mi>a</mi><mi>k</mi><mo>+</mo></msubsup><mi>sin</mi><mo>⁡</mo><mo stretchy="false">(</mo><mi>k</mi><mi>t</mi><mo stretchy="false">)</mo><mo>+</mo><msubsup><mi>a</mi><mi>k</mi><mo>−</mo></msubsup><mi>sin</mi><mo>⁡</mo><mo stretchy="false">(</mo><mo>−</mo><mi>k</mi><mi>t</mi><mo stretchy="false">)</mo></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mi>y</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo><munderover><mo>∑</mo><mrow><mi>k</mi><mo>=</mo><mn>1</mn></mrow><mn>4</mn></munderover><msubsup><mi>a</mi><mi>k</mi><mo>+</mo></msubsup><mi>cos</mi><mo>⁡</mo><mo stretchy="false">(</mo><mi>k</mi><mi>t</mi><mo stretchy="false">)</mo><mo>+</mo><msubsup><mi>a</mi><mi>k</mi><mo>−</mo></msubsup><mi>cos</mi><mo>⁡</mo><mo stretchy="false">(</mo><mo>−</mo><mi>k</mi><mi>t</mi><mo stretchy="false">)</mo></mrow></mstyle></mtd></mtr></mtable><annotation encoding="application/x-tex">\begin{gather*} 
z(t) = \sum_{k = 1}^{4} a_k^+e^{ikt} + a_k^-e^{-ikt} \\ 
x(t) = \sum_{k = 1}^{4} a_k^+\sin(kt) + a_k^-\sin(-kt)\\ 
y(t) = \sum_{k = 1}^{4} a_k^+\cos(kt) + a_k^-\cos(-kt)\\ 
\end{gather*}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:10.2097em;vertical-align:-4.8548em;"></span><span class="mord"><span class="mtable"><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:5.3548em;"><span style="top:-7.3548em;"><span class="pstrut" style="height:3.8011em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.04398em;">z</span><span class="mopen">(</span><span class="mord mathnormal">t</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.8011em;"><span style="top:-1.8479em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span><span class="mrel mtight">=</span><span class="mord mtight">1</span></span></span></span><span style="top:-3.05em;"><span class="pstrut" style="height:3.05em;"></span><span><span class="mop op-symbol large-op">∑</span></span></span><span style="top:-4.3em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">4</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.3021em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8213em;"><span style="top:-2.4086em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mbin mtight">+</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2914em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal">e</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8991em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">ik</span><span class="mord mathnormal mtight">t</span></span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8213em;"><span style="top:-2.4086em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mbin mtight">−</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2914em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal">e</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8991em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mathnormal mtight" style="margin-right:0.03148em;">ik</span><span class="mord mathnormal mtight">t</span></span></span></span></span></span></span></span></span></span></span><span style="top:-3.9516em;"><span class="pstrut" style="height:3.8011em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="mopen">(</span><span class="mord mathnormal">t</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.8011em;"><span style="top:-1.8479em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span><span class="mrel mtight">=</span><span class="mord mtight">1</span></span></span></span><span style="top:-3.05em;"><span class="pstrut" style="height:3.05em;"></span><span><span class="mop op-symbol large-op">∑</span></span></span><span style="top:-4.3em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">4</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.3021em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8213em;"><span style="top:-2.4086em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mbin mtight">+</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2914em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop">sin</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="mord mathnormal">t</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8213em;"><span style="top:-2.4086em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mbin mtight">−</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2914em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop">sin</span><span class="mopen">(</span><span class="mord">−</span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="mord mathnormal">t</span><span class="mclose">)</span></span></span><span style="top:-0.5484em;"><span class="pstrut" style="height:3.8011em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mopen">(</span><span class="mord mathnormal">t</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.8011em;"><span style="top:-1.8479em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span><span class="mrel mtight">=</span><span class="mord mtight">1</span></span></span></span><span style="top:-3.05em;"><span class="pstrut" style="height:3.05em;"></span><span><span class="mop op-symbol large-op">∑</span></span></span><span style="top:-4.3em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">4</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.3021em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8213em;"><span style="top:-2.4086em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mbin mtight">+</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2914em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop">cos</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="mord mathnormal">t</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8213em;"><span style="top:-2.4086em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mbin mtight">−</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2914em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop">cos</span><span class="mopen">(</span><span class="mord">−</span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="mord mathnormal">t</span><span class="mclose">)</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:4.8548em;"><span></span></span></span></span></span></span></span></span></span></span></span><br>
接下来就是画图了。代码如下：</p>
<pre data-role="codeBlock" data-info="matlab" class="language-matlab matlab"><code>clear<span class="token punctuation">;</span> clc<span class="token punctuation">;</span>

T <span class="token operator">=</span> <span class="token number">0</span><span class="token operator">:</span><span class="token number">.01</span><span class="token operator">:</span><span class="token number">2</span><span class="token operator">*</span><span class="token keyword keyword-pi">pi</span><span class="token punctuation">;</span>
X <span class="token operator">=</span> <span class="token number">12</span> <span class="token operator">*</span> <span class="token function">sin</span><span class="token punctuation">(</span>T<span class="token punctuation">)</span> <span class="token operator">-</span> <span class="token number">4</span> <span class="token operator">*</span> <span class="token function">sin</span><span class="token punctuation">(</span><span class="token number">3</span> <span class="token operator">*</span> T<span class="token punctuation">)</span><span class="token punctuation">;</span>
Y <span class="token operator">=</span> <span class="token number">13</span> <span class="token operator">*</span> <span class="token function">cos</span><span class="token punctuation">(</span>T<span class="token punctuation">)</span> <span class="token operator">-</span> <span class="token number">5</span> <span class="token operator">*</span> <span class="token function">cos</span><span class="token punctuation">(</span><span class="token number">2</span> <span class="token operator">*</span> T<span class="token punctuation">)</span> <span class="token operator">-</span> <span class="token number">2</span> <span class="token operator">*</span> <span class="token function">cos</span><span class="token punctuation">(</span><span class="token number">3</span> <span class="token operator">*</span> T<span class="token punctuation">)</span> <span class="token operator">-</span> <span class="token function">cos</span><span class="token punctuation">(</span><span class="token number">4</span> <span class="token operator">*</span> T<span class="token punctuation">)</span><span class="token punctuation">;</span>

a_x <span class="token operator">=</span> <span class="token punctuation">[</span><span class="token number">12</span><span class="token punctuation">,</span> <span class="token number">0</span><span class="token punctuation">,</span> <span class="token operator">-</span><span class="token number">4</span><span class="token punctuation">,</span> <span class="token number">0</span><span class="token punctuation">]</span><span class="token punctuation">;</span>
a_y <span class="token operator">=</span> <span class="token punctuation">[</span><span class="token number">13</span><span class="token punctuation">,</span> <span class="token operator">-</span><span class="token number">5</span><span class="token punctuation">,</span> <span class="token operator">-</span><span class="token number">2</span><span class="token punctuation">,</span> <span class="token operator">-</span><span class="token number">1</span><span class="token punctuation">]</span><span class="token punctuation">;</span>
a_p <span class="token operator">=</span> <span class="token punctuation">(</span>a_y <span class="token operator">+</span> a_x<span class="token punctuation">)</span> <span class="token operator">/</span> <span class="token number">2</span><span class="token punctuation">;</span>
a_n <span class="token operator">=</span> <span class="token punctuation">(</span>a_y <span class="token operator">-</span> a_x<span class="token punctuation">)</span> <span class="token operator">/</span> <span class="token number">2</span><span class="token punctuation">;</span>

writerObj <span class="token operator">=</span> <span class="token function">VideoWriter</span><span class="token punctuation">(</span><span class="token string">'heart.mp4'</span><span class="token punctuation">,</span> <span class="token string">'MPEG-4'</span><span class="token punctuation">)</span><span class="token punctuation">;</span>
writerObj<span class="token punctuation">.</span>FrameRate <span class="token operator">=</span> <span class="token number">15</span><span class="token punctuation">;</span>
<span class="token function">open</span><span class="token punctuation">(</span>writerObj<span class="token punctuation">)</span><span class="token punctuation">;</span>

figure<span class="token punctuation">;</span>
<span class="token keyword keyword-for">for</span> frame <span class="token operator">=</span> <span class="token number">1</span><span class="token operator">:</span><span class="token function">length</span><span class="token punctuation">(</span>T<span class="token punctuation">)</span>
    <span class="token function">plot</span><span class="token punctuation">(</span><span class="token function">X</span><span class="token punctuation">(</span><span class="token number">1</span><span class="token operator">:</span>frame<span class="token punctuation">)</span><span class="token punctuation">,</span> <span class="token function">Y</span><span class="token punctuation">(</span><span class="token number">1</span><span class="token operator">:</span>frame<span class="token punctuation">)</span><span class="token punctuation">,</span><span class="token string">'b'</span><span class="token punctuation">,</span> <span class="token string">'LineWidth'</span><span class="token punctuation">,</span> <span class="token number">2</span><span class="token punctuation">)</span><span class="token punctuation">;</span>hold on<span class="token punctuation">;</span>
    x <span class="token operator">=</span> <span class="token number">0</span><span class="token punctuation">;</span>
    y <span class="token operator">=</span> <span class="token number">0</span>
    <span class="token keyword keyword-for">for</span> <span class="token number">i</span> <span class="token operator">=</span> <span class="token number">1</span><span class="token operator">:</span><span class="token number">4</span>
        <span class="token function">xlim</span><span class="token punctuation">(</span><span class="token punctuation">[</span><span class="token operator">-</span><span class="token number">20</span><span class="token punctuation">,</span> <span class="token number">20</span><span class="token punctuation">]</span><span class="token punctuation">)</span><span class="token punctuation">;</span>
        <span class="token function">ylim</span><span class="token punctuation">(</span><span class="token punctuation">[</span><span class="token operator">-</span><span class="token number">20</span><span class="token punctuation">,</span> <span class="token number">15</span><span class="token punctuation">]</span><span class="token punctuation">)</span><span class="token punctuation">;</span>
        <span class="token function">plot</span><span class="token punctuation">(</span><span class="token punctuation">[</span>x<span class="token punctuation">,</span> x <span class="token operator">+</span> <span class="token function">a_p</span><span class="token punctuation">(</span><span class="token number">i</span><span class="token punctuation">)</span> <span class="token operator">*</span> <span class="token function">sin</span><span class="token punctuation">(</span><span class="token number">i</span> <span class="token operator">*</span> <span class="token function">T</span><span class="token punctuation">(</span>frame<span class="token punctuation">)</span><span class="token punctuation">)</span><span class="token punctuation">]</span><span class="token punctuation">,</span> <span class="token punctuation">[</span>y<span class="token punctuation">,</span> y <span class="token operator">+</span> <span class="token function">a_p</span><span class="token punctuation">(</span><span class="token number">i</span><span class="token punctuation">)</span> <span class="token operator">*</span> <span class="token function">cos</span><span class="token punctuation">(</span><span class="token number">i</span> <span class="token operator">*</span> <span class="token function">T</span><span class="token punctuation">(</span>frame<span class="token punctuation">)</span><span class="token punctuation">)</span><span class="token punctuation">]</span><span class="token punctuation">)</span><span class="token punctuation">;</span>hold on<span class="token punctuation">;</span>
        x <span class="token operator">=</span> x <span class="token operator">+</span> <span class="token function">a_p</span><span class="token punctuation">(</span><span class="token number">i</span><span class="token punctuation">)</span> <span class="token operator">*</span> <span class="token function">sin</span><span class="token punctuation">(</span><span class="token number">i</span> <span class="token operator">*</span> <span class="token function">T</span><span class="token punctuation">(</span>frame<span class="token punctuation">)</span><span class="token punctuation">)</span><span class="token punctuation">;</span>
        y <span class="token operator">=</span> y <span class="token operator">+</span> <span class="token function">a_p</span><span class="token punctuation">(</span><span class="token number">i</span><span class="token punctuation">)</span> <span class="token operator">*</span> <span class="token function">cos</span><span class="token punctuation">(</span><span class="token number">i</span> <span class="token operator">*</span> <span class="token function">T</span><span class="token punctuation">(</span>frame<span class="token punctuation">)</span><span class="token punctuation">)</span><span class="token punctuation">;</span>
        <span class="token function">plot</span><span class="token punctuation">(</span><span class="token punctuation">[</span>x<span class="token punctuation">,</span> x <span class="token operator">-</span> <span class="token function">a_n</span><span class="token punctuation">(</span><span class="token number">i</span><span class="token punctuation">)</span> <span class="token operator">*</span> <span class="token function">sin</span><span class="token punctuation">(</span><span class="token number">i</span> <span class="token operator">*</span> <span class="token function">T</span><span class="token punctuation">(</span>frame<span class="token punctuation">)</span><span class="token punctuation">)</span><span class="token punctuation">]</span><span class="token punctuation">,</span> <span class="token punctuation">[</span>y<span class="token punctuation">,</span> y <span class="token operator">+</span> <span class="token function">a_n</span><span class="token punctuation">(</span><span class="token number">i</span><span class="token punctuation">)</span> <span class="token operator">*</span> <span class="token function">cos</span><span class="token punctuation">(</span><span class="token number">i</span> <span class="token operator">*</span> <span class="token function">T</span><span class="token punctuation">(</span>frame<span class="token punctuation">)</span><span class="token punctuation">)</span><span class="token punctuation">]</span><span class="token punctuation">)</span><span class="token punctuation">;</span>hold on<span class="token punctuation">;</span>
        x <span class="token operator">=</span> x <span class="token operator">-</span> <span class="token function">a_n</span><span class="token punctuation">(</span><span class="token number">i</span><span class="token punctuation">)</span> <span class="token operator">*</span> <span class="token function">sin</span><span class="token punctuation">(</span><span class="token number">i</span> <span class="token operator">*</span> <span class="token function">T</span><span class="token punctuation">(</span>frame<span class="token punctuation">)</span><span class="token punctuation">)</span><span class="token punctuation">;</span>
        y <span class="token operator">=</span> y <span class="token operator">+</span> <span class="token function">a_n</span><span class="token punctuation">(</span><span class="token number">i</span><span class="token punctuation">)</span> <span class="token operator">*</span> <span class="token function">cos</span><span class="token punctuation">(</span><span class="token number">i</span> <span class="token operator">*</span> <span class="token function">T</span><span class="token punctuation">(</span>frame<span class="token punctuation">)</span><span class="token punctuation">)</span><span class="token punctuation">;</span>
    <span class="token keyword keyword-end">end</span>
    hold off<span class="token punctuation">;</span>
    img <span class="token operator">=</span> <span class="token function">getframe</span><span class="token punctuation">(</span>gcf<span class="token punctuation">)</span><span class="token punctuation">;</span>
    <span class="token function">writeVideo</span><span class="token punctuation">(</span>writerObj<span class="token punctuation">,</span> img<span class="token punctuation">)</span>
<span class="token keyword keyword-end">end</span>

<span class="token function">close</span><span class="token punctuation">(</span>writerObj<span class="token punctuation">)</span><span class="token punctuation">;</span>

<span class="token comment">% 作者：功夫螃蟹</span>
<span class="token comment">% 链接：https://www.zhihu.com/question/43309577/answer/3537040812</span>
<span class="token comment">% 来源：知乎</span>

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